of 1 25
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Bone!
By!
Ian Beardsley!
Copyright © January 4 2020!
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!
of 4 25
Part 1!
It would seem there is some possibility that as life goes from the fundamental framework to
form, so does the math that describes it. I make no attempt to understand why the
composition of life is aesthetically pleasing and how that came about, but I think it is due to the
need for function. The structure herein found arose when I was comparing biological life to
artificial intelligence. The case for bone was so interesting to me that I decided to proceed to
muscle and skin. It is because of this structure only revealing itself when comparing the
biological to AI, that I might suggest you can only speak about what life is relative to another
construct, like AI. Especially where awareness is concerned if we consider the Turing test."
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Bone
In my exploration of the connection between biological life and AI the most dynamic
component is that of bone. It affords us the opportunity to look at:!
Multiplying Binomials!
Completing The Square!
The Quadratic Formula!
Ratios!
Proportions!
The Golden Ratio!
The Square Root of Two!
The Harmonic Mean"
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Density of silicon is Si=2.33 grams per cubic centimeter.!
Density of germanium is Ge=5.323 grams per cubic centimeter.!
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.!
This is!
where !
Where HA is the mineral component of bone, Si is an AI semiconductor material and Ge is an
AI semiconductor material. This means!
!
The harmonic mean between Si and Ge is HA,…!
!
This is the sextic,…!
!
Which has a solution!
!
Where x=Si, and y=Ge. It can be solved with the online Wolfram Alpha computational engine.
But,…!
!
!
"
3
4
Si +
1
4
Ge ≈ HA
HA = Ca
5
(PO
4
)
3
OH
Si
HA
Si +
[
1 −
Si
HA
]
Ge = H A
2SiGe
Si + Ge
≈ HA
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
Ge
=
1
2 + 1
1
HA
2
Si
2
−
Ge
HA
2
Si +
[
Ge
HA
− 1
]
= 0
Si =
1
2
Ge
±
HA
Ge
HA
2
−
4Ge
HA
+ 4
Si = Ge − H A
of 7 25
!
!
!
!
!
!
Si
HA
Si +
[
1 −
Si
HA
]
Ge = H A
Si
2
HA
+ Ge −
Si
HA
Ge ≈ HA
1
HA
Si
2
−
Ge
HA
Si + Ge ≈ HA
1
HA
2
Si
2
−
Ge
HA
2
Si +
Ge
HA
≈ 1
1
HA
2
Si
2
−
Ge
HA
2
Si +
Ge
HA
− 1 ≈ 0
1
HA
2
Si
2
−
Ge
HA
2
Si +
[
Ge
HA
− 1
]
= 0
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!
!
We see that the square of the binomial is a quadratic where the third term is the square of one
half the middle coefficient. This gives us a method to solve quadratics called completing the
square:!
!
!
!
!
!
!
!
"
(x + a)(x + a) = x
2
+ 2a x + a
2
(x + a)
2
= x
2
+ 2a x + a
2
a x
2
+ bx + c = 0
a x
2
+ bx = − c
x
2
+
b
a
x = −
c
a
(
1
2
b
a
)
2
=
1
4
b
2
a
2
x
2
+
b
a
x +
1
4
b
2
a
2
= −
c
a
+
1
4
b
2
a
2
(
x +
1
2
b
a
)
2
=
b
2
− 4ac
4a
2
x +
b
2a
=
±
b
2
− 4ac
2a
x =
−b
±
b
2
− 4ac
2a
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!
!
!
!
!
!
!
!
!
!
!
"
1
HA
2
Si
2
−
Ge
HA
2
Si +
[
Ge
HA
− 1
]
= 0
x =
−b
±
b
2
− 4ac
2a
a =
a
HA
2
b = −
Ge
HA
2
c =
[
Ge
HA
− 1
]
b
2
− 4ac =
Ge
2
HA
4
− 4
1
HA
2
[
Ge
HA
− 1
]
=
Ge
2
HA
4
−
4Ge
HA
3
+
4
HA
2
=
1
HA
2
[
Ge
2
HA
2
−
4Ge
HA
+ 4
]
b
2
− 4ac =
1
HA
(
Ge
HA
− 2
)
2
x =
Ge
HA
2
±
1
HA
[
Ge
HA
− 2
]
2
HA
2
=
1
2
Ge
±
1
2
HA
[
Ge
HA
− 2
]
=
1
2
Ge
±
1
2
Ge − H A
Si =
1
2
Ge +
1
2
Ge − H A
Si = Ge − H A
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!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
Si ≈ Ge − H A
HA ≈
2SiGe
Si + Ge
Si ≈ Ge −
2SiGe
Si + Ge
(Si + Ge)Ge
Si + Ge
−
(Si + Ge)Si
Si + Ge
−
2SiGe
Si + Ge
= 0
Ge
2
− 2SiGe − Si
2
Si + Ge
= 0
x
2
− 2x y − y
2
= 0
x
2
− 2x y = y
2
x
2
− 2x y + y
2
= 2y
2
(x − y)
2
= 2y
2
x − y =
±
2y
x = y + 2y
x = y(1 + 2)
x
y
= 1 + 2
y
x
=
1
2 + 1
Si
Ge
≈
1
2 + 1
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A ratio is and a proportion is which means a is to b as b is to c.!
The Golden Ratio !
and. !
or !
!
!
!
!
!
!
!
!
a
b
a
b
=
b
c
(
Φ
)
a
b
=
b
c
a = b + c
ac = b
2
c =
b
2
a
a = b +
b
2
a
b
2
a
− a + b = 0
b
2
a
2
− 1 +
b
a
= 0
(
b
a
)
2
+
b
a
− 1 = 0
(
b
a
)
2
+
b
a
+
1
4
= 1 +
1
4
(
b
a
+
1
2
)
2
=
5
4
b
a
= −
1
2
±
5
2
b
a
=
5 − 1
2
a
b
=
5 + 1
2
ϕ =
5 − 1
2
Φ =
5 + 1
2
ϕ =
1
Φ
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The mineral component of bone hydroxyapatite (HA) is!
!
The organic component of bone is collagen which is!
!
We have!
!
!
!
!
%!
!
"
Ca
5
(PO
4
)
3
OH = 502.32
g
mol
C
57
H
91
N
19
O
16
= 1298.67
g
mol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
0.381966011
0.386795722
100 = 98.75
Si
Ge
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
Ge
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
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Part 2!
In comparing biological life to artificial intelligence I found the case for bone very compelling for
life as a mathematical construct. It is really quite interesting, if not aesthetically pleasing in its
structure. I covered that in my paper Life As A Mathematical Construct, here I revisit bone
using those findings in terms of the intermembral index which compares the forelimbs of
vertebrates to their hindlimbs. A ratio greater than one means the forelimbs are longer than the
hindlimbs and less than one the hindlimbs are longer. It is this ratio that tells paleontologists a
great deal about the manner of propulsion of a vertebrate."
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We begin by looking at hydroxyapatite (HA) the mineral component of bone and comparing it to
silicon (Si) and germanium (Ge) the most basic components of artificial intelligence (AI).!
Density of silicon is Si=2.33 grams per cubic centimeter.!
Density of germanium is Ge=5.323 grams per cubic centimeter.!
Density of hydroxyapatite is HA=3.00 grams per cubic centimeter.!
This is!
where !
This gives!
and. !
Which is the sextic!
!
Which has solution!
!
Because if y=Si, and x=Ge, we have!
!
!
!
!
!
3
4
Si +
1
4
Ge ≈ HA
HA = Ca
5
(PO
4
)
3
OH
Si
HA
Si +
[
1 −
Si
HA
]
Ge = H A
2SiGe
Si + Ge
≈ HA
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si
Ge
=
1
2 + 1
Si ≈ Ge − H A
HA ≈
2SiGe
Si + Ge
Ge
2
− 2SiGe − Si
2
Si + Ge
= 0
x
2
− 2x y − y
2
= 0
y
x
=
1
2 + 1
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Thus we see that if i is the intermembral index and Si and Ge subscripted with rho are the
densities of silicon and germanium, respectively. Thus,…!
!
!
And,…!
!
Now we introduce the organic component of bone, collagen:!
!
!
We have!
!
!
!
!
%!
!
!
By molar mass. Where, is the golden ratio conjugate, is recurrent throughout ratios in
vertabrates determined by bone lengths."
i ≈ 0.7
Si
ρ
Ge
ρ
≈
1
2 + 1
1
i
≈
1
2 + 1
+ 1
Ca
5
(PO
4
)
3
OH = 502.32
g
mol
C
57
H
91
N
19
O
16
= 1298.67
g
mol
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
= 0.386795722
ϕ = 0.618033989
1 − ϕ = 0.381966011
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
≈ (1 − ϕ)
0.381966011
0.386795722
100 = 98.75
Si
Ge
=
28.09
72.61
= 0.386861314 ≈ (1 − ϕ)
Si
Ge
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
ϕ
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Part 3!
I find 1/4 coupled with 3/4 is recurrent throughout Nature and that where there is (1/4, 3/4)
there is (1/3, 2/3) where 1/3 = 1- 2/3. An example of this would be in Mendelian genetics. 3/4
of the off spring express a recessive trait. This is because while the male has XY sex
chromosomes the female has XX sex chromosomes. Thus,…!
XY + XX = 3X + Y or 1Y: 3X=>1/3 and 1-1/3=2/3!
But we can say,…!
XY + XX = X + X + X + Y = 4 elements!
And,…!
!
And, 1-1/4=3/4.!
This is how (1/4, 3/4) is coupled with (1/3, 2/3)"
Y
X + X + X + Y
=
1
4
elem ents
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The Ancient Greek Elements"
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Air is about 25% oxygen gas (O2) by volume and 75% nitrogen gas (N2) by volume meaning
the molar mass of air as a mixture is:!
!
By density we find!
!
(Density of Ge is 5.323 g/cm^3 and of Si is 2.33 g/cm^3)!
is the mineral component of bone.!
This is the source of our equations:!
!
!
The density of HA is about 3.00 g per cubic centimeter.!
!
!
!
!
Thus if we consider!
!
Then we should consider sea water, or ocean:!
Thus we have,…!
0.25O
2
+ 0.75N
2
≈ air
0.25Ge + 0.75Si ≈ H A
HA = hydrox yapatite = Ca
5
(PO
4
)
3
OH
Si
HA
(Si) +
[
1 −
Si
HA
]
(Ge) ≈ HA
2(Si)(Ge)
Si + Ge
≈ HA
Si
HA
(Si) +
[
1 −
Si
HA
]
(Ge) ≈ HA
1
HA
2
Si
2
−
Ge
HA
2
Si +
[
Ge
HA
− 1
]
= 0
Si =
1
2
Ge
±
HA
Ge
HA
2
−
4Ge
HA
+ 4
Si ≈ Ge − HA
1
4
O
2
+
3
4
N
2
≈ air
of 20 25
!
!
!
The Ancient Greeks categorized Nature by 4 elements they called Earth, Air, Water, and Fire. In
a sense this was a way of saying solid, gas, liquid, energy. The Earth crust is highest in oxygen,
mainly because a lot of that is in water (hydrogen combined with oxygen). We have!
Oxygen (O) = 467,100 ppm!
Silicon (Si) = 276,900 ppm!
Aluminum (Al) = 80,700 ppm!
Iron (Fe) = 50,500 ppm!
If we say metal is Al +Fe = 131,200 and dirt is 276,900 then metal+dirt=408,100.!
Dirt= 276,900/408,100 = 67.85%!
Metal = 131200/408,100 = 32%!
Thus,…!
0.67(dirt) + 0.33(metal) = earth or!
!
Is (1/3, 2/3) by parts.!
!
Is (1/4, 3/4) by molar mass.!
!
For Sea or ocean water by molar mass.!
!
By density, where Ge and Si are mineral and sand components."
NaCl = 58.44 g/m ol an d H
2
O = 18.02 g/mol
1
3
NaCl ≈ H
2
O an d
2
3
H
2
O ≈ Carbon (C ) 12.01 g/mol
2
3
H
2
O + NaCl
NaCl
+
1
3
H
2
O
NaCl
≈ 1
1
3
metal +
2
3
dir t ≈ ear th
1
4
O
2
+
3
4
N
2
≈ air
2
3
H
2
O + NaCl
NaCl
+
1
3
H
2
O
NaCl
≈ 1
1
4
Ge +
3
4
Si = bone
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Why Take The Long Way Home?"
of 22 25
Why notice that the density of Si divided by the density of HA is about 3/4 and write!
!
And then use that with!
!
When it means we have to solve the sextic!
!
When we could have just noticed !
!
And then merely have to solve the quadratic!
!
The reason is that by first looking at the ratios of our densities instead of their differences we
have bone as part of the Ancient Greek elements and that takes us to archaeology. What is
Archaeology but the study of the decomposition of bone in its exposure to earth, air, and
water?!
I found it interesting that bone was appearing in the context of Earth, Air, and Water. I knew
that the oldest skeletons of humans and their ancient ancestors were unearthed in Africa, and
that Chinese archaeologists wanted to find skeletons in China just as old so they could show
they arose independently of any other people. The argument as to why they had not found
anything as old as in Africa was that Africa is drier so skeletons preserve longer. And, this
brings me to my point:!
The organic component of bone is collagen and decomposes early on. What is left is the
mineral component (hydroxylapatite, HA). The amount of bone in eight weeks that decays in
soil (Earth) is the same as the amount of bone that decays in air in two weeks (CAP, 1986) and
the primary factors in bone decomposition are soil, air, and water. Mendelian genetics are are
at the core of biological evolution and the study of that is the study of bone decomposition. I
need only present now, my earlier work. The mineral component (HA) decomposes in exposure
to water into calcium and phosphates.!
Thus in humid conditions bones can decompose in decades and in dry conditions can be
preserved for thousands of years. Francesco Berna, Alan Mathews, and Stephen Weiner report
(2004):!
We measured the ionic activity products at “steady-state” conditions and we identify a
recrystallization window between pH 7.6 and 8.1, which defines the conditions under which
bone crystals dissolve and reprecipitate as a more insoluble form of carbonated hydroxyl
apatite. As these conditions are common in nature, most fossil bones will not maintain their
original crystals with time.!
Si
HA
Si +
[
1 −
Si
HA
]
Ge = H A
2SiGe
Si + Ge
≈ HA
x
2
(x + y)
4
− x y(x + y)
4
+ 2x y
2
(x + y)
3
− 4x
2
y
2
(x + y)
2
= 0
Si = Ge − H A
x
2
− 2x y − y
2
= 0
of 23 25
Part 4!
We have said!
!
!
So,…!
Or…!
!
!
We have also said!
!
!
!
!
!
In order for Si and Ge to semiconductor they must be doped. Often the doping agents are
phosphorus (P) and boron (B). We find by atomic radius where SI_R=110pm, Ge_R=125pm,
P_R=100pm and B_r=85pm!
or,…. !
1
i
≈
1
2 + 1
+ 1 ≈ 2
Si
ρ
Ge
ρ
≈
1
2 + 1
1
i
≈
Si
ρ
Ge
ρ
+ 1
1
i
≈
Si
ρ
+ Ge
ρ
Ge
ρ
i ≈
Ge
ρ
Si
ρ
+ Ge
ρ
Si
M
Ge
M
≈
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
Ge
ρ
Si
ρ
+ Ge
ρ
≈
2
2
= sin
π
4
= cos
π
4
Si
M
Ge
M
≈ (1 − ϕ)
Si
M
≈ (1 − ϕ)Ge
M
Si
M
Ca
5
(PO
4
)
3
OH
C
57
H
91
N
19
O
16
Ge
M
P
R
+ B
R
Si
R
≈ Φ
Si
R
P
R
+ B
R
≈ ϕ
of 24 25
But if we take the arithmetic mean between the geometric mean of P_M and B_M and the
harmonic mean and divide by Si_M we find that,…!
!
Which yields,…!
!
Thus,…!
!
"
P
M
B
M
(P
M
+ B
M
) + 2P
M
B
M
2(P
M
+ B
M
)Si
M
≈ ϕ
2Si
R
P
R
+ B
R
Si
M
≈ P
M
P
M
+
2P
M
B
M
P
M
+ B
M
2Si
R
P
R
+ B
R
Si
M
≈
C
57
H
91
N
19
O
16
Ca
5
(PO
4
)
3
OH
[
P
M
B
M
+
2P
M
B
M
P
M
+ B
M
]
Ge
M
Ge
ρ
Si
ρ
+ Ge
ρ
≈
2
2
of 25 25
The Author
